1 exploration: parabolas and symmetry€¦ · 307 copyrigh dea earning c al igh eserved. copyright...

307Copyright © Big Ideas Learning, LLC

All rights reserved.

Copyright © Big Ideas Learning, LLC Integrated Mathematics III All rights reserved. Student Journal

49

8.7 Characteristics of Quadratic Functions For use with Exploration 8.7

Name _________________________________________________________ Date __________

Essential Question What type of symmetry does the graph of

( ) ( )2= − +f x a x h k have and how can you describe this symmetry?

Work with a partner.

a. Complete the table. Then use the values in the table to sketch the graph of the function

( ) 212 2 2f x x x= − − on graph paper.

b. Use the results in part (a) to identify the vertex of the parabola.

c. Find a vertical line on your graph paper so that when you fold the paper, the left portion of the graph coincides with the right portion of the graph. What is the equation of this line? How does it relate to the vertex?

d. Show that the vertex form ( ) ( )212 2 4f x x= − − is equivalent to the function

given in part (a).

1 EXPLORATION: Parabolas and Symmetry

x −2 −1 0 1 2

f (x)

x 3 4 5 6

f (x)

x

y

4

6

2

−4

−6

−2

4 62−2−4−6

x

46

2

y

4

6

2

−4

−6

−2

−2−4−6

308Copyright © Big Ideas Learning, LLCAll rights reserved.

Integrated Mathematics III Copyright © Big Ideas Learning, LLC Student Journal All rights reserved. 50

8.7 Characteristics of Quadratic Functions (continued)

Name _________________________________________________________ Date _________

Work with a partner. Repeat Exploration 1 for the function given by ( ) ( )221 1

3 32 3 3 6.f x x x x= − + + = − − +

Communicate Your Answer

3. What type of symmetry does the graph of ( ) ( )2f x a x h k= − + have and how can you describe this symmetry?

4. Describe the symmetry of each graph. Then use a graphing calculator to verify your answer.

a. ( ) ( )21 4f x x= − − + b. ( ) ( )21 2f x x= + − c. ( ) ( )22 3 1f x x= − +

d. ( ) ( )212 2f x x= + e. ( ) 22 3f x x= − + f. ( ) ( )23 5 2f x x= − +

2 EXPLORATION: Parabolas and Symmetry

x −2 −1 0 1 2

f (x)

x 3 4 5 6

f (x)

x

y

4

6

2

−4

−6

−2

4 62−2−4−6




51

8.7 For use after Lesson 8.7

Name _________________________________________________________ Date __________

In your own words, write the meaning of each vocabulary term.

axis of symmetry

standard form

minimum value

maximum value

intercept form

Core Concepts

Properties of the Graph of ( ) 2f x ax bx c= + + 2 , 0y ax bx c a= + + > 2 , 0y ax bx c a= + + <

• The parabola opens up when 0a > and open down when 0.a <

• The graph is narrower than the graph of ( ) 2 when 1f x x a= > and wider when 1.a <

• The axis of symmetry is 2bxa

= − and the vertex is , .2 2b bfa a

− −

• The y-intercept is c. So, the point ( )0, c is on the parabola.

Notes:

x

y

x = –

(0, c)

b2a

x

y

(0, c)

x = – b2a




51

8.7 Notetaking with Vocabulary For use after Lesson 8.7

Name _________________________________________________________ Date __________


axis of symmetry

standard form

minimum value

maximum value

intercept form

Core Concepts






− −


Notes:

x

y

x = –

(0, c)

b2a

x

y

(0, c)

x = – b2a




51

8.7 For use after Lesson 8.7

Name _________________________________________________________ Date __________


axis of symmetry

standard form

minimum value

maximum value

intercept form

Core Concepts






− −


Notes:

x

y

x = –

(0, c)

b2a

x

y

(0, c)

x = – b2a



8.7 Notetaking with Vocabulary (continued)

Name _________________________________________________________ Date _________

Minimum and Maximum Values For the quadratic function ( ) 2 ,f x ax bx c= + + the y-coordinate of the vertex is the minimum value of the function when 0a > and the maximum value when 0.a < 0a > 0a <

• Minimum value: 2bfa

−

• Domain: All real numbers

• Range: 2by fa

≥ −

• Decreasing to the left of 2bxa

= −

• Increasing to the right of 2bxa

= −

Notes:

Properties of the Graph of ( ) ( )( )f x a x p x q= − −

• Because ( ) ( )0 and 0,f p f q= = p and q are the x-intercepts of the graph of the function.

• The axis of symmetry is halfway between ( ) ( ), 0 and , 0 .p q

So, the axis of symmetry is .2

p qx +=

• The parabola opens up when 0a > and opens down when 0.a <

Notes:

• Maximum value: 2bfa

−


• Range: 2by fa

≤ −

• Increasing to the left of 2bxa

= −

• Decreasing to the right of 2bxa

= −

x

y

minimum

increasingdecreasing

x = – b2a

x

y

maximum

increasing decreasing

x = – b2a

x

y

(q, 0)

(p, 0)

x =

y = a(x – p)(x – q)

p + q2




51

8.7 Notetaking with Vocabulary For use after Lesson 8.7

Name _________________________________________________________ Date __________


axis of symmetry

standard form

minimum value

maximum value

intercept form

Core Concepts






− −


Notes:

x

y

x = –

(0, c)

b2a

x

y

(0, c)

x = – b2a

Practice



8.7

Name _________________________________________________________ Date _________



−


• Range: 2by fa

≥ −


= −


= −

Notes:





p qx +=


Notes:


−


• Range: 2by fa

≤ −


= −


= −

x

y

minimum


x = – b2a

x

y

maximum


x = – b2a

x

y

(q, 0)

(p, 0)

x =

y = a(x – p)(x – q)

p + q2



8.7

Name _________________________________________________________ Date _________



−


• Range: 2by fa

≥ −


= −


= −

Notes:





p qx +=


Notes:


−


• Range: 2by fa

≤ −


= −


= −

x

y

minimum


x = – b2a

x

y

maximum


x = – b2a

x

y

(q, 0)

(p, 0)

x =

y = a(x – p)(x – q)

p + q2


Integrated Mathematics III Copyright © Big Ideas Learning, LLC

Student Journal All rights reserved. 52


Name _________________________________________________________ Date _________



−


• Range: 2by fa

≥ −


= −

• Increasing to the right of bxa

= −

Notes:





p qx +=


Notes:


−


• Range: 2by fa

≤ −


= −


= −

x

y

minimum


x = – b2a

x

y

maximum


x = – b2a

x

y

(q, 0)

(p, 0)

x =

y = a(x – p)(x – q)

p + q2

Worked-Out Examples

Example #1

Graph the function. Label the vertex and axis of symmetry.

Example #2

Graph the function. Label the vertex and axis of symmetry

80 Integrated Mathematics III Copyright © Big Ideas Learning, LLC Worked-Out Solutions All rights reserved.

Chapter 2

4. Identify the constants a = 1, h = −4, and k = 0. Plot the vertex (h, k) = (−4, 0) and draw the axis of symmetry x = −4. Evaluate the function for two values of x.

x = −3: h(−3) = (−3 + 4)2 = 1

x = −2: h(−2) = (−2 + 4)2 = 4

Plot the points (−3, 1), (−2, 4), and their refl ections in the axis of symmetry. Draw a parabola through the plotted points.

x

4

6

2

−2−6

y

(−4, 0)

x = −4


x = −1: g(−1) = (−1 + 3)2 + 5 = 9

x = −2: g(−2) = (−2 + 3)2 + 5 = 6


x

4

6

8

2

−2−4−6

y

(−3, 5)

x = −3

6. Identify the constants a = 1, h = 7, and k = −1. Plot the

vertex (h, k) = (7, −1) and draw the axis of symmetry x = 7. Evaluate the function for two values of x. x = 5: y = (5 − 7)2 − 1 = 3 x = 6: y = (6 − 7)2 − 1 = 0 Plot the points (5, 3), (6, 0), and their refl ections in the axis of symmetry. Draw a parabola through the plotted points.

x

4

6

2

2 4 10 12

−2

y

(7, −1)

x = 7

7. Identify the constants a = −4, h = 2, and k = 4. Plot the vertex (h, k) = (2, 4) and draw the axis of symmetry x = 2. Evaluate the function for two values of x.

x = 4: y = −4(4 − 2)2 + 4 = −20

x = 3: y = −4(3 − 2)2 + 4 = 0

Plot the points (4, −20), (3, 0), and their refl ections in the axis of symmetry. Draw a parabola through the plotted points.

x

4

6

2

4

−2

−4

y

(2, 4)

x = 2

8. Identify the constants a = 2, h = −1, and k = −3. Plot the vertex (h, k) = (−1, −3) and draw the axis of symmetry x = −1. Evaluate the function for two values of x.

x = 1: g(1) = 2(1 + 1)2 − 3= 5

x = 0: g(0) = 2(0 + 1)2 − 3 = −1

Plot the points (1, 5), (0, −1), and their refl ections in the axis of symmetry. Draw a parabola through the plotted points.

x

4

2

2−4

y

(−1, −3)

x = −1

9. Identify the constants a = −2, h = 1, and k = −5. Plot the vertex (h, k) = (1, −5) and draw the axis of symmetry x = 1. Evaluate the function for two values of x.

x = 3: f (3) = −2(3 − 1)2 − 5= −13

x = 2: f (2) = −2(2 − 1)2 − 5 = −7

Plot the points (3, −13), (2, −7), and their refl ections in the axis of symmetry. Draw a parabola through the plotted points.

x2 4−2

−2

−4

−6

−8

y

(1,

−5)

x = 1

80 Integrated Mathematics III Copyright © Big Ideas Learning, LLC

Worked-Out Solutions


Chapter 2


x = −3: h(−3) = (−3 + 4)2 = 1

x = −2: h(−2) = (−2 + 4)2 = 4


x

4

6

2

−2−6

y

(−4, 0)

x = −4


x = −1: g(−1) = (−1 + 3)2 + 5 = 9

x = −2: g(−2) = (−2 + 3)2 + 5 = 6


x

4

6

8

2

−2−4−6

y

(−3, 5)

x = −3

6. Identify the constants a = 1, h = 7, and k = −1. Plot the vertex (h, k) = (7, −1) and draw the axis of symmetry x = 7. Evaluate the function for two values of x.

x = 5: y = (5 − 7)2 − 1 = 3

x = 6: y = (6 − 7)2 − 1 = 0

Plot the points (5, 3), (6, 0), and their refl ections in the axis of symmetry. Draw a parabola through the plotted points.

x

4

6

2

2 4 10 12

−2

y

(7, −1)

x = 7


x = 4: y = −4(4 − 2)2 + 4 = −20

x = 3: y = −4(3 − 2)2 + 4 = 0


x

4

6

2

4

−2

−4

y

(2, 4)

x = 2


x = 1: g(1) = 2(1 + 1)2 − 3= 5

x = 0: g(0) = 2(0 + 1)2 − 3 = −1

Plot the points (1, 5), (0, −1), and their refl ections in the axis of symmetry. Draw a parabola through the plotted points.

x

4

2

2−4

y

(−1, −3)

x = −1


x = 3: f (3) = −2(3 − 1)2 − 5= −13

x = 2: f (2) = −2(2 − 1)2 − 5 = −7


x2 4−2

−2

−4

−6

−8

y

(1, −5)

x = 1


Chapter 2

15. C; The axis of symmetry is x = 3.

16. D; The axis of symmetry is x = −4.

17. B; The axis of symmetry is x = −1.

18. A; The axis of symmetry is x = 2.

19.

x

4

6

2

6−2−4

y

(4, −1)

(2, 3)(3, 2)

x = 2

20.

x

6

2

2−2−4−6

−2

−4

−6

y

(−3, −3)(−4, −2)

(−5, 1)

x = −3

21. Identify the coeffi cients a = 1, b = 2, and c = 1. Because a > 0, the parabola opens up. Find the vertex. First calculate the x-coordinate.

x = − b —

2a = −

2 —

2(1) = −1

Then fi nd the y-coordinate of the vertex.

y = (−1)2 + 2(−1) + 1 = 0

So, the vertex is (−1, 0). Plot this point. Draw the axis of symmetry. Identify the y-intercept, c, which is 1. Plot the point (0, 1) and its refl ection in the axis of symmetry, (−2, 1). Evaluate the function for another value of x, such as x = 1.

y = 12 + 2(1) + 1 = 4

Plot the point (1, 4) and its refl ection in the axis of symmetry, (−3, 4). Draw a parabola through the plotted points.

x

4

6

8

2

42−2−4

y

(−1, 0)

x = −1

22. Identify the coeffi cients a = 3, b = 6, and c = 4. Because a > 0, the parabola opens up. Find the vertex.the x-coordinate.

x = − b —

2a = −6 —

2(3) = 1


y = 3(1)2 − 6(1) + 4 = 1

So, the vertex is (1, 1). Plot this point. Draw the axis of symmetry. Identify the y-intercept, c, which is 4.(2, 4). Evaluate the function for another value

as x = 3.

y = 3(3)2 − 6(3) + 4 = 13


x

4

2

2

y

(1, 1)

x = 1

Plot the point (0, 4) and its refl ection in the axis of symmetry,




Name _________________________________________________________ Date _________



−


• Range: 2by fa

≥ −


= −


= −

Notes:





p qx +=


Notes:


−


• Range: 2by fa

≤ −


= −


= −

x

y

minimum


x = – b2a

x

y

maximum


x = – b2a

x

y

(q, 0)

(p, 0)

x =

y = a(x – p)(x – q)

p + q2




Name _________________________________________________________ Date _________



−


• Range: 2by fa

≥ −


= −


= −

Notes:





p qx +=


Notes:


−


• Range: 2by fa

≤ −


= −


= −

x

y

minimum


x = – b2a

x

y

maximum


x = – b2a

x

y

(q, 0)

(p, 0)

x =

y = a(x – p)(x – q)

p + q2

Practice (continued)

( )27 1y x − −

First calculate

of x, such

2y x x= +− 3 6

=

4




53

8.7

Name _________________________________________________________ Date __________

Extra Practice In Exercises 1–3, graph the function. Label the vertex and axis of symmetry. Find the minimum or maximum value of the function. Describe the domain and range of the function, and where the function is increasing and decreasing.

1. ( ) ( )21f x x= + 2. ( )22 4 5y x= − − − 3. ( ) 232 3 1t x x x= − −

In Exercises 4 and 5, graph the function. Label the x-intercept(s), vertex, and axis of symmetry.

4. ( ) ( )( )4 4 3f x x x= + − 5. ( ) ( )7 6f x x x= − −

6. A softball player hits a ball whose path is modeled by ( ) 20.0005 0.2127 3,f x x x= − + + where x is the distance from home plate (in feet) and y is the height of the ball above the ground (in feet). What is the highest point this ball will reach? If the ball was hit to center field which has an 8 foot fence located 410 feet from home plate, was this hit a home run? Explain.

x

y

x

y

x

y

x

y

x

y

Practice A


Chapter 2


x = −3: h(−3) = (−3 + 4)2 = 1

x = −2: h(−2) = (−2 + 4)2 = 4


x

4

6

2

−2−6

y

(−4, 0)

x = −4


x = −1: g(−1) = (−1 + 3)2 + 5 = 9

x = −2: g(−2) = (−2 + 3)2 + 5 = 6


x

4

6

8

2

−2−4−6

y

(−3, 5)

x = −3

6. Identify the constants a = 1, h = 7, and k = −1. Plot the vertex (h, k) = (7, −1) and draw the axis of symmetry x = 7. Evaluate the function for two values of x.

x = 5: y = (5 − 7)2 − 1 = 3

x = 6: y = (6 − 7)2 − 1 = 0

Plot the points (5, 3), (6, 0), and their refl ections in the axis of symmetry. Draw a parabola through the plotted points.

x

4

6

2

2 4 10 12

−2

y

(7, −1)

x = 7


x = 4: y = −4(4 − 2)2 + 4 = −20

x = 3: y = −4(3 − 2)2 + 4 = 0


x

4

6

2

4

−2

−4

y

(2, 4)

x = 2


x = 1: g(1) = 2(1 + 1)2 − 3= 5

x = 0: g(0) = 2(0 + 1)2 − 3 − Plot the points (1, 5), (0, −1), and their refl ections in the axis

of symmetry. Draw a parabola through the plotted points.

x

4

2

2−4

y

(−1, −3)

x = −1


x = 3: f (3) = −2(3 − 1)2 − 5= −13

x = 2: f (2) = −2(2 − 1)2 − 5 = −7


x2 4−2

−2

−4

−6

−8

y

(1, −5)

x = 1


Chapter 2





19.

x

4

6

2

6−2−4

y

(4, −1)

(2, 3)(3, 2)

x = 2

20.

x

6

2

2−2−4−6

−2

−4

−6

y

(−3, −3)(−4, −2)

(−5, 1)

x = −3


x = − b —

2a = −

2 —

2(1) = −1


y = (−1)2 + 2(−1) + 1 = 0


y = 12 + 2(1) + 1 = 4


x

4

6

8

2

42−2−4

y

(−1, 0)

x = −1


x = − b —

2a = −6 —

2(3) = 1


y = 3(1)2 − 6(1) + 4 = 1

So, the vertex is (1, 1). Plot this point. Draw the axis of symmetry. Identify the y-intercept, c, which is 4. Plot the point (0, 4) and its refl ection in the axis of symmetry,

x, such as x = 3.

y = 3(3)2 − 6(3) + 4 = 13


x

4

2

2

y

(1, 1)

x = 1




53


Name _________________________________________________________ Date __________


1. ( ) ( )21f x x= + 2. ( )22 4 5y x= − − − 3. ( ) 232 3 1t x x x= − −


4. ( ) ( )( )4 4 3f x x x= + − 5. ( ) ( )7 6f x x x= − −


x

y

x

y

x

y

x

y

x

y

Practice A




53

8.7

Name _________________________________________________________ Date __________


1. ( ) ( )21f x x= + 2. ( )22 4 5y x= − − − 3. ( ) 232 3 1t x x x= − −


4. ( ) ( )( )4 4 3f x x x= + − 5. ( ) ( )7 6f x x x= − −


x

y

x

y

x

y

x

y

x

y

Practice A




53


Name _________________________________________________________ Date __________


1. ( ) ( )21f x x= + 2. ( )22 4 5y x= − − − 3. ( ) 232 3 1t x x x= − −


4. ( ) ( )( )4 4 3f x x x= + − 5. ( ) ( )7 6f x x x= − −


x

y

x

y

x

y

x

y

x

y

Practice A




53


Name _________________________________________________________ Date __________


1. ( ) ( )21f x x= + 2. 3. ( ) 232 3 1t x x x= − −


4. ( ) ( )( )4 4 3f x x x= + − 5. ( ) ( )7 6f x x x= − −


x

y

x

y

x

y

x

y

x

y

Practice A


Chapter 2





19.

x

4

6

2

6−2−4

y

(4, −1)

(2, 3)(3, 2)

x = 2

20.

x

6

2

2−2−4−6

−2

−4

−6

y

(−3, −3)(−4, −2)

(−5, 1)

x = −3


x = − b —

2a = −

2 —

2(1) = −1


y = (−1)2 + 2(−1) + 1 = 0


y = 12 + 2(1) + 1 = 4


x

4

6

8

2

42−2−4

y

(−1, 0)

x = −1


x = − b —

2a = −6 —

2(3) = 1


y = 3(1)2 − 6(1) + 4 = 1

So, the vertex is (1, 1). Plot this point. Draw the axis of symmetry. Identify the y-intercept, c, which is 4. Plot the point (0, 4) and its refl ection in the axis of symmetry, (2, 4). Evaluate the function for another value of x, such as x = 3.

y = 3(3)2 − 6(3) + 4 = 13


x

4

2

2

y

(1, 1)

x = 1

Practice (continued)

( )22 4 5y x= − − −


Integrated Mathematics III Copyright © Big Ideas Learning, LLC Resources by Chapter All rights reserved. 58

2.6 Practice B

Name _________________________________________________________ Date _________

In Exercises 1–12, graph the function. Label the vertex and axis of symmetry.

1. ( ) ( )23 2 4f x x= − − − 2. ( ) ( )23 1 5f x x= + +

3. ( ) ( )212 3 2g x x= − + + 4. ( ) ( )21

2 2 1h x x= − −

5. ( )20.6 2y x= − 6. ( ) 20.25 1f x x= −

7. 2 8y x= − + 8. 27 2y x= +

9. 21.5 6 3y x x= − + 10. ( ) 20.5 3 1f x x x= + −

11. 252 5 1y x x= − + 12. ( ) 23

2 6 4f x x x= − − −

13. A quadratic function is decreasing to the left of 3x = and increasing to the right of 3.x = Will the vertex be the highest or lowest point on the graph of the parabola? Explain.

14. The graph of which function has the same axis of symmetry as the graph of 22 8 3?y x x= − + Explain your reasoning.

A. 24 16 5y x x= − + − B. 22 8 7y x x= + +

C. 23 6 7y x x= − + D. 26 10 1y x x= − + −

In Exercises 15–18, find the minimum or maximum value of the function. Describe the domain and range of the function, and where the function is increasing and decreasing.

15. 23 12y x= + 16. 2 6y x x= − −

17. 213 2 3y x x= − − + 18. ( ) 21

2 3 7f x x x= + +

19. The height of a bridge is given by 23 ,y x x= − + where y is the height of the bridge (in miles) and x is the number of miles from the base of the bridge.

a. How far from the base of the bridge does the maximum height occur?

b. What is the maximum height of the bridge?

Practice B

1 exploration: parabolas and symmetry€¦ · 307 copyrigh dea earning c al igh eserved. copyright...

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